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United States Patent |
6,014,419
|
Hu
|
January 11, 2000
|
CT cone beam scanner with fast and complete data acquistion and accurate
and efficient regional reconstruction
Abstract
A method is provided for reconstructing an image of an object, for use in
an imaging system wherein a detector is mounted for measuring radiation
emanating in a cone beam of rays which converge at a focal point. The
method includes the step of establishing relative movement between the
cone beam focal point and the object along a composite scan path,
comprising primary and supplementary scan path components, the
supplementary path usefully comprising a linear or helical path. The
method further includes acquiring a set of cone beam data of the object
with the detector during movement along the supplementary scan path
component, computing a set of values of an intermediate function from the
cone beam data, each of said computed values having an associated location
defined by a prespecified point grid lying on the detector plane, and
employing each of the computed values in a back-projection operation to
determine the value of a reconstruction function, for use in forming an
image of the object. The method also includes a strategy of imaging
longitudinally-unbounded object section by section
Inventors:
|
Hu; Hui (1030 Ridge Rd., Waukesha, WI 53186)
|
Appl. No.:
|
966155 |
Filed:
|
November 7, 1997 |
Current U.S. Class: |
378/4; 378/901 |
Intern'l Class: |
A61B 006/03 |
Field of Search: |
378/4,15,901
|
References Cited
U.S. Patent Documents
5170439 | Dec., 1992 | Zeng et al. | 382/6.
|
5663995 | Sep., 1997 | Hu | 378/15.
|
5706325 | Jan., 1998 | Hu | 378/4.
|
5784481 | Jul., 1998 | Hu | 382/131.
|
5926521 | Jul., 1999 | Tam | 378/4.
|
Primary Examiner: Bruce; David Vernon
Attorney, Agent or Firm: Skarsten Law Offices S.C.
Claims
What is claimed is:
1. In an imaging system wherein a detector is mounted for measuring
radiation emanating in a cone beam of rays which converge at a focal
point, a method for reconstructing an image of an object comprising the
steps of:
establishing relative movement between said cone beam focal point and said
object along a composite scan path, comprising primary and supplementary
scan path components;
acquiring a set of cone beam data of said object with said detector during
movement along said supplementary scan path component, said cone beam data
associated with said supplementary scan path to be used only for deriving
information which cannot be derived from other cone beam data that is
associated with the primary scan path;
computing a set of values of an intermediate function from said cone beam
data, each of said computed values having an associated location defined
by a prespecified point grid lying on the detector plane; and
employing each of said computed values to compute the supplementary
component of a reconstruction function of said object.
2. The method of claim 1 wherein said object comprises a
longitudinally-unbounded object, and said method comprises the further
steps of:
defining a scan field of view relative to said imaging system;
determining error propagation distance in the direction along the normal of
a plane containing the primary scan path;
modifying the definition of the scan field of view, according to said error
propagation distance; and
generating the reconstruction within the modified field of view from the
composite scan path.
3. The method of claim 2 wherein:
repeated supplementary scans are made at different selected locations along
a Z-axis positioned with respect to said object until an entire
longitudinal extent of the longitudinally-unbounded object is imaged.
4. The method of claim 2 wherein:
repeated primary scans are made at different selected locations along a
Z-axis positioned with respect to said object until an entire region of
interest of the object is imaged.
5. The method of claim 4 wherein:
the maximum longitudinal displacement between adjacent primary scans, along
said Z-axis, is chosen so that within said object no gap exists between
the modified fields of view of said adjacent primary scans.
6. The method of claim 1 wherein:
said supplementary scan path component comprises at least a linear
component.
7. The method of claim 1 wherein:
said primary scan path component comprises at least a circular component.
8. The method of claim 1 wherein:
said value of said reconstruction function is computed by means of
interpolation of values of said intermediate function.
9. The method of claim 1 wherein:
said step of computing the values of said intermediate function comprises
summing weighted cone beam data only along the lines representing those
planes not intersecting the primary scan path.
10. The method of claim 1 wherein:
said supplementary scan path component comprises at least a helical
component.
11. In an imaging system wherein a detector is mounted for measuring
radiation emanating in a cone beam of rays which converge at a focal
point, a method for reconstructing an image of an object comprising the
steps of:
establishing relative movement between said cone beam focal point and said
object along a composite scan path, comprising primary and supplementary
scan path components;
acquiring a set of cone beam data of said object with said detector during
movement along said supplementary scan path component;
computing a set of values of an intermediate function from said cone beam
data, each of said computed values having an associated location defined
by a prespecified point grid lying on the detector plane, said values of
said intermediate function being computed on the detector plane on a grid
whose sampling spacing is substantially greater than that of the detector
elements projected on the detector plane; and
employing each of said computed values to compute the supplementary
component of a reconstruction function of said object.
12. In an imaging system wherein a detector is mounted for measuring
radiation emanating in a cone beam of rays which converse at a focal
point, a method for reconstructing an image of an object comprising the
steps of:
computing at least one part of the said reconstruction on a sparser
three-dimensional image grid, which, compared with a full-size image grid,
has a larger spacing between points in the x, y, and/or z directions;
combining thumbnail reconstructions of all said computed parts;
generating a combined full size reconstruction of said computed parts from
the combined thumbnail reconstructions by interpolation; and
combining the interpolated full size reconstruction with the full-size
reconstructions of other parts to produce the complete full-size
reconstruction.
13. In an imaging system wherein a detector is mounted for measuring
radiation emanating in a cone beam of rays which converge at a focal
point, a method for reconstructing an image of the object comprising the
steps of:
establishing relative movement between said cone beam focal point and said
object along a composite scan path comprising primary and curved
supplementary components;
acquiring a set of cone beam data of said object with said detector during
movement along said curved supplementary scan path; and
deriving an intermediate function from said cone beam data for use in
forming the supplementary component of the reconstruction function of said
object.
14. The method of claim 13 wherein:
said method includes acquiring a second set of cone beam data of said
object during movement along said primary scan path, generating a primary
component reconstruction function from said second set of cone beam data,
and combining said primary component reconstruction function with said
supplementary component reconstruction function to provide an image of
said object.
15. The method of claim 14 wherein said method includes the steps of:
computing at least one part of the said reconstruction on a sparser
three-dimensional image grid, which, compared with a full-size image grid,
has a larger spacing between points in the x, y, and/or z directions;
combining thumbnail reconstructions of all said computed parts;
generating a combined full size reconstruction of said computed parts from
the combined thumbnail reconstructions by interpolation; and
combining the interpolated full size reconstruction with the full-size
reconstructions of other parts to produce the complete full-size
reconstruction.
16. The method of claim 14 wherein:
said reconstruction method includes a filtering step applied to at least
some parts of the said reconstruction in selected relation to the
backprojection step.
17. The method of claim 14 wherein said object comprises a
longitudinally-unbounded object, and said method comprises the further
steps of:
defining a scan field of view relative to said imaging system;
determining error propagation distance in the direction along the normal of
a plane containing the primary scan path
modifying the definition of the scan field of view, according to said error
propagation distance; and
generating the reconstruction within the modified field of view from the
composite scan path.
18. The method of claim 19 wherein:
the maximum longitudinal displacement between adjacent primary scans, along
said Z-axis is chosen so that within said object no gap exists between the
modified fields of view of said adjacent primary scans.
19. The method of claim 17 wherein:
repeated primary scans are made at different selected locations, along a
Z-axis positioned with respect to said object until an entire region of
interest of the object is imaged.
20. The method of claim 17 wherein:
repeated supplementary scans are made at different selected locations along
a Z-axis positioned with respect to said object until an entire
longitudinal extent of the longitudinally-unbounded object is imaged.
21. The method of claim 13 wherein:
said supplementary scan path component comprises at least a helical
component.
22. The method of claim 13 wherein:
said primary scan path component comprises at least a circular component.
23. The method of claim 13 wherein:
said value of said reconstruction function is computed by means of
interpolation of values of said intermediate function.
24. The method of claim 13 wherein:
the values of said intermediate function are computed on a detector plane
on a grid whose sampling spacing is substantially greater than that of the
detector elements projected on the detector plane.
25. The method of claim 13 wherein said deriving step comprises:
computing a set of values of an intermediate function from said cone beam
data, each of said computed values having an associated location defined
by a prespecified point grid lying on the detector plane; and
employing each of said computed values to compute the supplementary
component of reconstruction of said object.
26. The method of claim 13 wherein:
said cone beam data associated with movement along the supplementary scan
path is used only for deriving information which cannot be derived from
other cone beam data that is associated with the primary scan path.
27. The method of claim 13 wherein:
said step of computing the values of said intermediate function comprises
summing weighted cone beam data only along the lines representing those
planes not intersecting the primary scan path.
Description
BACKGROUND OF THE INVENTION
The present invention generally pertains to cone beam computed tomography
(CT) imaging apparatus and method. More particularly, the invention
pertains to such apparatus and method that acquires and processes cone
beam projection data acquired along a trajectory comprising a circular or
other primary scan path (i.e., orbit) supplemented by a helical or other
supplementary scan path.
Cone beam CT imaging has developed as an important technique in
constructing a three-dimensional CT image. According to such technique, an
X-ray source irradiates the object with conical shaped X-rays while
traversing a prescribed scan path or trajectory, to project an image of
the object, in the form of cone beam X-ray data, onto an array of
two-dimensional detector elements. The detector elements acquire or
receive the projected cone beam data, which is then processed to provide
the reconstructed image of the object.
Scan path is an essential consideration in cone beam imaging. Different
scan paths represent different data measurement procedures and call for
different data processing algorithms (reconstruction algorithms) to
produce the reconstructed images. Developing accurate, efficient and
robust reconstruction algorithms for the scan paths of practical interest
has been the focus of many research groups. As a prerequisite for high
fidelity (exact) reconstruction, the scan path employed should provide
sufficient cone beam data measurements.
The reconstruction algorithm for generating a primary part of the
reconstructed function from circular path cone beam CT was given by
Feldkamp et al, "Practical Cone-beam Algorithm", J. Opt. Soc. Am., pp.
612-619 (1984). The algorithm for generating the entire portion of the
reconstructed function that can be derived from circular path cone beam CT
was recently given by U.S. Pat. No. 5,400,255, issued Mar. 21, 1995, to
Hui Hu, the inventor herein. However, it is well known that the circular
scan path is likely to provide insufficient cone beam data and may
generate erroneous results.
Various scanning geometries (paths) have been developed to ensure that
sufficient data is acquired. In one such geometry, the scan path comprises
a circular path in combination with a linear path, which is orthogonal to
the plane containing the circular path. Various algorithms are currently
available for use in processing cone beam data acquired by scanning along
a combined circle-and-line path and constructing an image therefrom.
However, some of such algorithms, such as set forth in an article by H.
Kudo and T. Saito, entitled, "Derivation and implementation of a cone-beam
reconstruction algorithm for non-planar orbits", IEEE Trans. Med. Imag.
vol. 13 pp. 196-211 (1994) require excessive data processing resources.
Other of such algorithms, such as set forth in an article by G Zheng and
G. Gullberg entitled, "A cone beam tomography algorithm for orthogonal
circle-and-line orbit", Phys. Med. Biol., vol. 37(4) pp. 563-577 (1992)
and in U.S. Pat. No. 5,170,439 have been found to be inaccurate.
More recently, an reconstruction algorithm has been developed by the
inventor for generating, from the linear scanned data, a portion of the
reconstructed function supplementary to the primary portion derivable from
the circular scan. Thus, this supplementary portion is then additively
combined with the primary portion which is derived from the circular scan
in accordance to U.S. Pat. No. 5,400,255 to provide a complete
reconstruction of the function of the object. While this technique has
provided significant benefits in terms of the reconstruction accuracy, it
has been found that a substantial amount of processing effort is still
required, in order to derive the linear scan portion of the reconstructed
function. It would be desirable to significantly reduce the data
processing load by improving the efficiency of the technique.
The circle-and-line scan path is of great practical interest, since it can
be readily implemented by rotating the scanner or the object around a
circle in the circular scan and by translating the object along the axis
of rotation in the linear scan. However, since no rotation is allowed
during the linear scan, the time it takes to completely stop the rotation
before the linear scan and to reestablish the rotation for the sequential
circular scan after the linear scan is too long for some applications,
especially when using the circle-and-line scan repeatedly. To eliminate he
lengthy switching time and therefore increase the overall data acquisition
speed, the present invention proposes a new scan path, i.e., the
circle-and-helix scan path. It would be desirable to develop a
reconstruction algorithm for this scan path.
For most applications in medicine and some applications in industry, the
longitudinal extent of the object to be imaged exceeds the length which
can be scanned by the scanner in one scan. Such an object is referred to
as a longitudinally-unbounded object. One practical consideration in cone
beam CT system development is how to image the longitudinally-unbounded
object when only a portion of it is of interest or can be imaged in one
scan due to the limited detector extent. Most of the methods developed
cannot meet this challenge. It would be desirable to be able to develop a
strategy for exact reconstruction of the longitudinally-unbounded object
through a series of regional scans and reconstructions.
SUMMARY OF THE INVENTION
In a CT imaging system comprising a source of cone beam radiation and a
two-dimensional array of detector elements which are selectively
positioned with respect to an object, a new and improved cone beam scan
and reconstruction technique is provided.
The technique includes the step of establishing relative movement between
the cone beam focal point and the object along a composite scan path
comprising a circular orbit or other trajectory which lies in a single
plane, supplemented by a linear, helical or other scan path which is not
confined to the plane. These two scan path components are called the
primary orbit and the supplementary orbit respectively. The cone beam
source irradiates the object during such movement to project cone beam
data onto the two-dimensional detector, the projected data comprising a
primary data set and a supplementary data set acquired from the primary
and supplementary orbits, respectively. Two important embodiments, i.e.,
the circle-and-line and circle-and-helix scan path, are explicitly
discussed hereinafter.
The technique includes new cone beam CT reconstruction algorithms. These
reconstruction algorithms are based on the idea of decomposing the
function to be reconstructed into two components: 1) the primary
component, which can be derived from the primary orbit; and 2) the
supplementary component, which is the remaining part of the function to be
reconstructed. The algorithms propose to compute the primary and
supplementary components of the reconstructed function respectively from
the primary and supplementary cone-beam data.
The technique also includes a new method of computing the supplementary
component of the reconstructed function from the supplementary data set.
The method comprises 1) computing the values of an intermediate function
on a set of points on a detector plane; and 2) back-projecting the
intermediate function to produce the supplementary component of the
reconstruction. This technique further includes adopting a special sparse
grid on which the intermediate function is computed, and/or adopting a
special sparse three-dimensional grid on which a part of the reconstructed
function is initially computed.
The technique further includes a strategy for imaging a
longitudinally-unbounded object. The method comprises 1) identifying a
region for each scan where an error-free reconstruction can be obtained,
and 2) combining multiple error-free reconstructions from multiple scans
to generate an exact reconstruction of the longitudinally-unbounded object
over the entire region-of-interest.
OBJECTS OF THE INVENTION
An object of the invention is to provide a cone beam CT method and
apparatus for high speed acquisition of a cone beam projection data set
that is sufficient for an exact reconstruction.
Another object is to improve accuracy (i.e., fidelity) and efficiency
(i.e., speed) in generating the reconstruction of an object.
Another object is to extend the usefulness and robustness of a cone beam CT
system of the above type to image a longitudinally-unbounded object.
These and other objects of the invention will become more readily apparent
from the ensuing specification, taken together with the accompanying
drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic diagram illustrating principal elements of a CT cone
beam imaging system and a circle-and-line scan path.
FIG. 2 is a schematic diagram illustrating principal elements of a CT cone
beam imaging system and a circle-and-helix scan path.
FIG. 3 is a perspective view further illustrating a conventional CT imaging
system for use in implementing some embodiments of the invention.
FIG. 4 is a view showing a cone beam imaging arrangement with associated
coordinate systems and spatial parameters imposed thereon for use in
further illustrating some embodiments of the invention.
FIG. 5 is a block diagram demonstrating an implementation of some
embodiments of the invention.
FIG. 6 is a schematic diagram illustrating an embodiment of the invention
in connection with a longitudinally unbounded object.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring to FIG. 1, there are shown the principal components of a
cone-beam CT imaging system. A cone beam source 12 is positioned to
irradiate the object to be imaged 10, and to thereby project a conical
shaped beam onto a two-dimensional detector array 16, comprising a matrix
array of discrete detector elements (not shown in detail). The cone-beam
projection measurements represent, in analog form, the number of photons
that penetrate the object along the lines connecting the cone beam focal
point 12 and the respective detector elements. Such data is coupled to a
Data Acquisition System (DAS) 20, which converts analog data from the
respective detector elements into digital form for subsequent processing.
The digitized projection data is coupled to an image reconstruction
processor system 22, which operates on the projection data to reconstruct
an image representation of the object 10. The reconstructed images may be
presented in viewable form, for example, by means of an image display 24.
FIG. 1 further shows a circular orbit of motion 26 for the cone beam source
12 around the object 10, such orbit lying in a single plane called the
mid-plane 28. In one arrangement, detector array 16 is constrained to move
with source 12, while object 10 remains still therebetween. Cone-beam
projection data is acquired by detector array 16 for successive positions
of the cone beam source 12 as the source 12 traverses the circular orbit
26. A z-axis 30 represents the axis of rotation, which passes through the
center of the circle, the point C, in orthogonal relationship with
mid-plane 28.
FIG. 1 further shows that the cone beam focal point 12 can be moved along a
linear orbit 34, which is not confined to the mid-plane.
The circle-and-line scan path can be readily implemented by a conventional
CT system. Referring to FIG. 3, there is shown a conventional CT system
substantially comprising a gantry 38 and a table 40. The table 40, which
is slideable upon base 46, moves a patient 44 linearly, along the z-axis.
Thus, table 40 can be operated to position a selected section 50 of the
patient within the bore 42, so that images can be taken therethrough.
Referring further to FIGS. 3 and 1 together, there is shown the source 12
and detector array 16 mounted on rotatable gantry 38, on opposing sides of
the bore 42. Accordingly, the circular orbit 26 may be established by
rotation of gantry 38. The linear orbit 34 may be established by linear
movement of the patient table 40, while source 12 and detector array 16
remain stationary.
The circle-and-line orbit is one embodiment of the present invention.
However, for the circle-and-line orbit, no rotation is allowed during the
linear scan. Thus, it is necessary to stop the gantry rotation after the
circular scan in order to start the linear scan, and to reestablish the
gantry rotation after the linear scan in order to start the next circular
scan. For some applications, the time spent on switching between the
circular and linear scans could represent a very significant portion of
the overall data acquisition time, especially when switching back and
forth repeatedly in multiple scans.
As another embodiment of the present invention, a new scan orbit comprising
a circle 26 and a helix 88 is proposed and shown in FIG. 2. The
circle-and-helix scan orbit can also be readily implemented by a
conventional CT system by a relative rotational movement and a relative
linear movement between the object and the gantry, as described above in
connection with FIG. 3. Since the rotational movement is required in both
circular and helical scans, the lengthy switching time required for
stopping and reestablishing the rotational movement is eliminated. This
represents a significant improvement in the overall data acquisition
speed.
The circle-and-line scan path and circle-and-helix scan path represent two
examples of a class of general composite scan paths. Each composite scan
path in this class comprises a primary orbit lying in a single plane 28
and a supplementary orbit not confined to the plane 28 containing the
primary orbit. The plane containing the primary orbit is referred to as
the mid-plane. It is to be understood that the primary orbit, besides the
circular orbit discussed herein, also includes an elliptical or other form
in other embodiments, provided that such orbit lies entirely in a single
plane. Similarly, the supplementary orbit, besides the linear or helical
scan orbit discussed herein, also includes other scan paths not confined
to the mid-plane.
The cone-beam projection data sets acquired from the primary and
supplementary scan orbits are referred to respectively as the primary and
supplementary data sets. More particularly, the cone-beam projection data
sets acquired from the circular, linear and helical scan orbits are
referred to respectively as the circularly, linearly and helically scanned
data sets.
As another embodiment of the present invention, new cone beam CT
reconstruction algorithms are proposed for the primary-and-supplementary
orbit in general, and for the circle-and-line scan path and
circle-and-helix scan path in particular. These reconstruction algorithms
are based on the idea of decomposing the function to be reconstructed into
two components: 1) the primary component, which can be derived from the
primary orbit; and 2) the supplementary component, which is the remaining
part of the function to be reconstructed. The algorithms propose to
compute the primary and supplementary components of the reconstructed
function respectively from the primary and supplementary data sets.
FIG. 4 illustrates the physical meanings of some elements and parameters
used in the presented invention. The x, y, and z axes in FIG. 4 represent
a Cartesian coordinate system that is fixed relative to the object to be
imaged. The z axis is along the axis of rotation 30 and the x and y axes
lie in the mid-plane 28. The point C is the origin of this fixed
coordinate system. The position of a point is expressed as r or (x, y, z)
in this Cartesian coordinate system. In FIG. 4, the point S represents the
focal point of a cone beam projection. The point O is the perpendicular
projection point of the point S onto the axis of rotation 30. Since cone
beam source 12 both rotates and translates, it is useful to provide an
additional coordinate system that moves with the cone beam source. The
orthogonal vectors of this moving coordinate system are (x', y', z'),
where x' and z' are directed along the line SO and the axis of rotation
30, i.e., the z-axis, respectively. The point O is the origin of this
moving coordinate system. The rotation movement of the cone beam source is
characterized by its rotational angle, .phi., relative to the fixed
coordinate system. The translational movement of the cone beam source is
characterized by its z elevation, z.sub.0, relative to the mid-plane 28.
The distance from O to S is denoted as d.
A special plane, referred to as the detector plane, is defined as a plane
perpendicular to the line SO. Without losing generality, the detector
plane 52 discussed herein and shown in FIG. 4 is chosen to contain the
z-axis. Any physical detector arrangement can be converted to this
detector plane by means of a mapping process. Thus, the position on the
detector plane, identified by the coordinates (Y,Z), also corresponds to
the physical position of the detector element. Therefore, a set of cone
beam projections acquired from the primary and supplementary scan orbit
can be characterized as P.sub..phi. (Y,Z) and P.sub.z0 (Y,Z) respectively,
where the position of the cone beam source on the primary orbit is
characterized by its rotational angle .phi., while the position of the
cone beam source on the supplementary orbit is characterized by its
distance z.sub.0 to the mid-plane. It will be readily apparent that
z.sub.0 will have a non-zero value only when cone beam source 12 is
positioned along the supplementary orbit above or below the mid-plane 28.
Both the primary and supplementary cone beam data sets are weighted as
follows to generate weighted projection data:
##EQU1##
U.S. Pat. No. 5,400,255, issued Mar. 21, 1995 to Hui Hu, the inventor
herein, teaches that for the circle orbit, any function to be
reconstructed f(r) can be decomposed into the following three terms:
f(r)=f.sub.C.sbsb.0 (r)+f.sub.C.sbsb.1 (r)+f.sub.L (r) (2)
The f.sub.C.sbsb.0 (r) term, computed from the circularly scanned data set,
corresponds to the Feldkamp reconstruction, which is formulated as the
following two steps:
1) p.sub..phi. (Y,Z)=.intg.dY'P.sub..phi. (Y',Z)h(Y--Y') (3a)
##EQU2##
where, h(Y) is the kernel of the ramp filter.
U.S. Pat. No. 5,400,255 teaches how to compute f.sub.C.sbsb.1 (r) from the
circular scan. It can be summarized as the following two steps:
##EQU3##
In accordance with the present invention, the f.sub.C.sbsb.0 (r) and
f.sub.C.sbsb.1 (r) terms combined form the primary component of the
function to be reconstructed. The f.sub.L (r) term represents the
supplementary component of the function to be reconstructed. As will be
discussed hereinafter, decomposing the function to be reconstructed into
several terms (such as shown in Equation 2) enables development of a
term-specific technique to further improve the accuracy and efficiency of
reconstruction of each term and therefore of the overall reconstruction.
U.S. Pat. No. 5,400,255 proposes to estimate f.sub.L (r) when only the
circularly scanned data is available. A related technique, described
hereinafter, teaches how to accurately generating f.sub.L (r) from the
linearly scanned data set. Such technique can be summarized as the
following three steps:
1) A line integral, .SIGMA..sub.z.sbsb.0 (l,.dwnarw.), may be computed by
summing the weighted projected data P.sub.z0 (Y,Z) at each (Y,Z) position
along the line L as follows:
.SIGMA..sub.z.sbsb.0 (l,.dwnarw.)=.intg..intg.dYdZP.sub.z.sbsb.0
(Y,Z).delta.(Y sin.dwnarw.+Z cos.dwnarw.-1) (5a)
Referring to FIG. 4, the line L represents the intersection line between a
plane W containing the source (i.e., the focal point) S 12 and the
detector plane 52. The line L is characterized by coordinates
(l,.dwnarw.), where l is the distance from the origin O to the line, and
.dwnarw. is the angle the normal of the line L makes with the z-axis 30.
For each cone beam source position z.sub.0 along the linear scan orbit,
computation of the line integration, .SIGMA..sub.z.sbsb.0 (l,.dwnarw.), is
carried out only for those lines which correspond to those planes W
intersecting the object but not intersecting the circular scan orbit.
These lines can be characterized by the following equation:
2lz.sub.0 cos.dwnarw.+z.sub.0.sup.2 cos.sup.2 .dwnarw.-d.sup.2 sin.sup.2
.dwnarw.>0 (5b)
To reflect this selective computation, the following selecting function
w.sub.z.sbsb.0 (l,.dwnarw.) is introduced:
##EQU4##
2) An intermediate function, H.sub.z.sbsb.0 (l,.dwnarw.), is computed
based on the line integrals .SIGMA..sub.z.sbsb.0 (l,.dwnarw.) and the
selecting function w.sub.z.sbsb.0 (l,.dwnarw.) as follows:
##EQU5##
3) The linear component, f.sub.L (r), could be determined by integrating
H.sub.z.sbsb.0 (l,.dwnarw.) over z.sub.0 and .dwnarw. as follows:
##EQU6##
Though computing f.sub.L (r) using Equations 5 significantly improves the
reconstruction accuracy, it still requires a substantial amount of
processing effort. This is because the integral operation
.intg..sub.0.sup..pi. d.dwnarw. H.sub.z.sbsb.0 (l,.dwnarw.) must be
carried out in the backprojection step (Equation 5e) for all the
reconstruction points, r or (x, y, z), and for each z.sub.0 along the
linear scan, which is very time-consuming.
In accordance with the present invention, the second and third steps of the
previous invention are modified as follows:
2) A new intermediate function, B.sub.z.sbsb.0 (Y,Z), is computed based on
the line integrals .SIGMA..sub.z.sbsb.0 (l,.dwnarw.) and the selecting
function w.sub.z.sbsb.0 (l,.dwnarw.) as follows:
##EQU7##
3) The linear component, f.sub.L (r), is determined by backprojecting
B.sub.z.sbsb.0 (Y, Z) as follows:
##EQU8##
In this new formulation, the new intermediate function B.sub.z.sbsb.0
(Y,Z), representing the integral .intg..sub.0.sup..pi. d.dwnarw.
H.sub.z.sbsb.0 (l,.dwnarw.), is pre-calculated on a set of points (Y,Z) on
the detector plane. This calculation is carried out once for each z.sub.0
along the linear scan prior to the backprojection step. In the
backprojection step of Equation 6e, the value of the intermediate function
B.sub.z.sbsb.0 (Y,Z) at any point, (Y.sub.0,Z.sub.0), is computed by
interpolation of those values pre-calculated on the grid (Y,Z), instead of
evaluating
##EQU9##
repeatedly as suggested by Equation 5e. The grid (Y,Z) on which the
intermediate function B.sub.z.sbsb.0 (Y,Z) is pre-calculated is referred
to as the pre-calculation grid.
The new formulation takes the integral
##EQU10##
out of the computationally intensive backprojection operation, which loops
through all the reconstruction points, r or (x,y, z). As a result, the
processing efficiency is significantly improved.
As an important embodiment of the present invention, the reconstruction
algorithm for the circle-and-helix scan orbit shown in FIG. 2 is proposed
in a similar form as follows: 1) compute the f.sub.C.sbsb.0 (r) and
f.sub.C.sbsb.1 (r) reconstruction using Equations 3 and 4 respectively; 2)
compute the supplementary component, f.sub.L (r), from the helically
scanned data using the following formulae (Equations 7a, below); and 3)
combine the three terms to form a complete reconstruction f(r).
Similar to the reconstruction from the linear scan, the reconstruction of
the supplementary component, f.sub.L (r), from the helically scanned data
also consists of three steps:
1) A line integral, .SIGMA..sub.z.sbsb.0 (l,.dwnarw.), may be computed by
summing the weighted projected data P.sub.z0 (Y,Z) at each (Y,Z) position
along the line L as follows:
.SIGMA..sub.z.sbsb.0 (l,.dwnarw.)=.intg..intg.dYdZP.sub.z.sbsb.0
(Y,Z).delta.(Y sin+Z cos.dwnarw.-1) (7a)
where the selecting function w.sub.z.sbsb.0 (l,.dwnarw.) is defined as:
##EQU11##
2) A new intermediate function, B.sub.z.sbsb.0 (Y,Z), is computed based on
the line integrals .SIGMA..sub.z.sbsb.0 (l,.dwnarw.) and the selecting
function w.sub.z.sbsb.0 (l,.dwnarw.) as follows:
B.sub.z0 (Y,Z)=
##EQU12##
3) The linear component, f'.sub.L (r), is determined by backprojecting
B.sub.z.sbsb.0 (Y,Z) as follows:
##EQU13##
In Equations 7, P.sub.z0 (Y,Z) represents the weighted cone beam projection
acquired from a helical scan, and k is a proportion factor so that z.sub.0
=k.phi..
The cone beam reconstruction algorithms for the circle-and-line or
circle-and-helix orbit are explicitly given in Equations 1-7. It is to be
understood that the reconstruction algorithms for other primary orbits,
such as the elliptical orbit, and/or for other supplementary orbits can be
derived by the coordinate transform method. Conventional examples of using
such method are set forth in the following references: B. Horn, "Fan-beam
reconstruction method," in Proc. IEEE, vol. 67, pp. 1616-1623 (1979); G.
Gullberg and G. Zeng, "A cone-beam filtered backprojection reconstruction
algorithm for cardiac single photon emission computed tomography," IEEE
Trans. Med. Imag., vol. 11, no. 1, pp. 91-101 (1992); G. Wang, T. Lin, and
P. Cheng, "A derivation-free noncircular fan-beam reconstruction formula",
IEEE Trans. image processing, vol. 2, no. 4, pp. 543-547 (1992).
In a modification of the present invention, which further significantly
improves the computational efficiency, it has been recognized that the
supplementary component, f.sub.L (r), while changing rapidly along the z
direction, varies slowly in the x and y directions.
Without utilizing this property of the f.sub.L (r) component, it would be
assumed that the pre-calculation grid would be similar to the matrix of
the detector elements projected on the detector plane. However, because of
the property of the f.sub.L (r) component, a much sparser sampling of
B.sub.z.sbsb.0 (Y,Z) along the Y direction should be sufficient. Thus, one
can use a pre-calculation grid distinctly different from the matrix of the
detector elements projected on the detector plane. More specifically, in
the Y direction, the sampling spacing of the pre-calculation grid can be
substantially greater (e.g., 10 times greater) than that of the detector
elements projected on the detector plane without losing any information.
On the other hand, in the Z direction, the sampling spacing of the
pre-calculation grid is comparable to or slightly higher than that of the
detector elements projected on the detector plane to maintain the high
frequency content in the z direction. Increasing the Y sampling spacing by
a factor of N will reduce the time for computing Equation 6d or 7d by
roughly a factor of N. The factor for optimized image x-y pitch may be on
the order of 10.
Applying a similar idea to the backprojection, the f.sub.L (r) in Equation
Se, 6e, or 7e can first be computed on a sparser three-dimensional point
grid in image space, which, compared with full-size image grid, has a
larger spacing between points in both the x and y directions. (Image space
has rectilinear coordinates (x,y,z), where the z axis coincides with the
Z-axis of the detector plane.) After the backprojection and before f(r) is
formed by combining three terms as described above, the values of f.sub.L
(r) on a full-size image grid can be computed by interpolation of those on
the sparser image grid. This sparser 3-D reconstruction is referred to as
a thumbnail reconstruction of the full size reconstruction. Increasing the
x-y pitch by a factor of N will reduce the computation time for Equation
5e, 6e, or 7e by roughly a factor of N.sup.2. The factor N for optimized
image x-y pitch may be on the order of 2-4.
It is to be noted that the two sparse sampling approaches mentioned above
are general and can be used to improve the reconstruction efficiency of
any function which has high frequency contents in one direction and low
frequency contents in other directions. For example, the approach directed
to sparse sampling in image space can be used to improve the computation
efficiency of f.sub.C1 (r) reconstruction (Equation 4b).
In a modification of the present invention, the concept of filtration can
be used to improve reconstruction accuracy and stability. The filter can
be applied to all or some of the components of the said reconstruction
(Equation 2), prior to or after the backprojection step. The filter can be
either shift-invariant or shift-variant, along one or multiple directions.
More specifically, the filter along the z direction may be necessary.
Referring to FIG. 5, there are shown certain operations performed in image
processing system 22, in accordance with the above equations.
Supplementary data P.sub.z0 (Y,Z) and primary data P.sub..phi. (Y,Z) are
coupled to a weighting process block 54 in accordance with Equation 1.
The supplementary data is then processed separately in block 56. Referring
further to FIG. 5, there is shown block 56 including a computation block
60, which operates to provide line integrals .SIGMA..sub.z0 (l,.dwnarw.)
by summing the weighted projection data P.sub.z0 (Y,Z) in accordance with
Equation 5a or 7a above. Only the integrals along those line which
correspond to planes not intersecting the primary orbit are computed.
These lines are selected so that its spatial parameters
(Z.sub.0,.dwnarw.,l) makes the selection function w.sub.z.sbsb.0
(l,.dwnarw.) non-zero, in accordance with Equation 5c or 7c. In block 62,
the an intermediate function B.sub.z0 (Y,Z) from the line integrals is
computed in accordance with Equation 6d or 7d. The in intermediate
function is then interpolated and backprojected in blocks 64 and 66 to
provide f.sub.L (r), in accordance with Equation 6e or 7e.
FIG. 5 further shows that the primary data P.sub..phi. (Y,Z) is sent to
primary data process block 58, which computes image reconstruction
functions f.sub.c1 (r) and f.sub.c0 (r) therefrom to provide the function
f.sub.c (r).
The functions f.sub.L (r) and f.sub.c (r) are respectively coupled to a
summing device 68 to provide the function f(r). It will be understood that
certain conventional functions performed by processor 22 are not necessary
for understanding the invention, and are accordingly not shown.
FIG. 6 shows the geometry of the cone-beam CT scanner with limited detector
z extent. The region within the heavy solid line in FIG. 6 indicates the
cross-section of the measurable region for a given source location (S1).
The measurable region is primarily determined by the detector extent and
the imaging geometry. The scan field of view (FOV) is defined as the
overlapping region of all measurable regions for all source locations in
the circular scan. Thus, the scan FOV is defined by a cylinder ended with
a cone on each end, whose axes coincide with the axis of rotation. The
cross-section of the scan FOV is darkly shaded in FIG. 6. The radius of
the cylinder is determined by the detector in-plane extent and the imaging
geometry. The top and bottom cones are determined by the upper and lower
cone angles, .beta..sub.max and .beta..sub.min shown in FIG. 6.
Missing some projection measurements due to the limited detector z extent
will introduce errors. In general, these errors may propagate inwardly
from the top and the bottom and contaminate the reconstruction in the top
and bottom layers of the scan FOV. The depth of the contaminated layers,
referred to as the contamination depth, are determined by the error
propagating distance along the Z direction. Different operators employed
by different reconstruction algorithms will result in different
contamination depths.
The rod-like object raises a challenging problem for cone beam CT scan and
reconstruction in general. However, this invention provides a solution to
this problem.
The contamination depth for each term in Equation 1 was analyzed by
examining the. 2, 3, and 4 or 5-7. It is concluded from Equation 2 that
the first term, that is, the Feldkamp reconstruction has a zero
contamination depth since no operation causes the errors to propagate in
the Z direction. It is concluded from Equation 3 that the contamination
depth of the second term is half of the detector Z cell pitch since a
difference operator (Equation 3a) is applied in the Z direction.
Furthermore, it is concluded from numerical analyses of Equation 5-7 that
the contamination depth of the third term is one detector Z cell pitch.
Once the contamination depth for each term is quantified, the strategy for
imaging the rod-like object becomes clear. Since the maximum depth of the
contaminated layer for this hybrid algorithm is one detector Z cell pitch,
these contaminated layers can be excluded by slightly modifying the
definition of the scan FOV so that its cone ends move inwardly by one
detector Z cell pitch at both ends.
Thus, for the primary-plus-supplementary scan path cone-beam CT using the
reconstruction algorithms proposed in this invention, the reconstruction
within the modified scan FOV does not require the missing measurements due
to the limited detector z extent. Therefore, exact regional reconstruction
of the longitudinally-unbounded object can be achieved within the modified
scan FOV.
Furthermore, the longitudinally-unbounded (rod-like) object can be imaged
section by section, where each section is imaged by one
primary-plus-supplementary scan. To maximize the volume coverage speed of
contiguous exams, the maximum longitudinal displacement between the
adjacent primary scans is chosen so that within the object to be imaged no
gap exists between the adjacent modified scan FOVs.
While FIG. 1 shows a planer detector array 16, it will be understood that
another embodiment of the invention could employ a different type of
detector, such as an array of detector cells lying along a curved surface,
or even a single detector cell or linear array of detector cells acquiring
the cone beam projection data sequentially.
Obviously, many other modifications and variations of the present invention
are possible in light of the above teachings. It is therefore to be
understood that within the scope of the disclosed concept, the invention
may be practiced otherwise than as specifically described.
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